Speaker：Kaiyuan Gu (Peking University)

Time：2026-10-29 15: 00

Location：Conference Room S106 at Experiment Building at Haiyun Campus

Abstract:

The Uniform Boundedness Conjecture (UBC) predicts that given an abelian variety A of dimension g defined over a number field K, the number of rational torsion points on A is bounded above solely in terms of g and K. While known for elliptic curves by the work of Mazur, Kamienny, and Merel, the conjecture remains widely open in higher dimensions. Recently, Looper and Yap proved its geometric analogue over function fields of characteristic zero, building on the method of Hindry--Silverman and Deligne's Arakelov inequality.

In this talk, I will present a new proof of the Geometric Uniform Boundedness Conjecture (GUBC), yielding explicit bounds with polynomial dependence on the base field. Our approach is inspired by Vojta's method of Diophantine approximation and incorporates new ideas from modern Arakelov geometry. The same method also give a proof of the Lang--Silverman conjecture over function fields of characteristic 0. Over number fields, we prove both conjectures conditionally on a higher-dimensional analogue of the Szpiro conjecture.

As a further application, our method yields bounds on the torsion subgroup of the Néron--Severi group of a smooth projective variety over C, a result that may be of independent interest.

This talk is based on a joint work with Ziyang Gao.